transforming a texture from one object to another
If your new model is in quads and is the same size, sits in the same place as the original then you should be able to bake the texture from the original to the new model as long as you unnwrap your new model and set up a texture to bake to that is high enough resolution to accommodate the pixels of the original texture. You would use the 'bake selected from active' option in the render panel bake tab.